Lemma 106.11.4. Let f : \mathcal{X} \to \mathcal{Y} and h : \mathcal{U} \to \mathcal{X} be morphisms of algebraic stacks. Assume that \mathcal{Y} is locally Noetherian, that f is locally of finite type and quasi-separated, that h is of finite type, and that the image of |h| : |\mathcal{U}| \to |\mathcal{X}| is dense in |\mathcal{X}|. If given any 2-commutative diagram
\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_-u \ar[d]_ j & \mathcal{U} \ar[r]_ h & \mathcal{X} \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[rr]^-y \ar@{..>}[rru] & & \mathcal{Y} }
where A is a discrete valuation ring with field of fractions K and \gamma : y \circ j \to f \circ h \circ u, the category of dotted arrows is either empty or a setoid with exactly one isomorphism class, then f is separated.
Proof.
We have to prove \Delta is a proper morphism. Assume first that \Delta is separated. Then we may apply Lemma 106.11.3 to the morphisms \mathcal{U} \to \mathcal{X} and \Delta : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}. Observe that \Delta is quasi-compact as f is quasi-separated. Of course \Delta is locally of finite type (true for any diagonal morphism, see Morphisms of Stacks, Lemma 101.3.3). Finally, suppose given a 2-commutative diagram
\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_-u \ar[d]_ j & \mathcal{U} \ar[r]_ h & \mathcal{X} \ar[d]^\Delta \\ \mathop{\mathrm{Spec}}(A) \ar[rr]^-y \ar@{..>}[rru] & & \mathcal{X} \times _\mathcal {Y} \mathcal{X} }
where A is a discrete valuation ring with field of fractions K and \gamma : y \circ j \to \Delta \circ h \circ u. By Morphisms of Stacks, Lemma 101.41.1 and the assumption in the lemma we find there exists a unique dotted arrow. This proves the last assumption of Lemma 106.11.3 holds and the result follows.
In the general case, it suffices to prove \Delta is separated since then we'll be back in the previous case. In fact, we claim that the assumptions of the lemma hold for
\mathcal{U} \to \mathcal{X} \quad \text{and}\quad \Delta : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}
Namely, since \Delta is representable by algebraic spaces, the category of dotted arrows for a diagram as in the previous paragraph is a setoid (see for example Morphisms of Stacks, Lemma 101.39.2). The argument in the preceding paragraph shows these categories are either empty or have one isomorphism class. Thus \Delta is separated.
\square
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